Arsenal’s early-season anxieties about the battle for a Champions League place seem to be easing, and it is largely because the excellent Sanchez has already found his feet following his transfer from Barcelona.
Arsenal were also able to hand a comeback to England winger Theo Walcott, who made his first appearance of the season as a second-half substitute after 10 months on the sidelines with a serious knee injury.
The game at the Emirates Stadium, where Arsenal manager Arsene Wenger named an unchanged team, quickly fell into a predictable pattern.
By the 10th minute, Arsenal had had close to 80 percent of possession. By the 30th minute, they had had more than four times as many shots on goal as Burnley.
By the 40th, the Lancashire side were already hugely grateful to goalkeeper Tom Heaton, after he made a series of fine saves.
Two of those stops were from Chilean forward Sanchez, who has now scored four times in two games after claiming a double in last weekend’s 2-0 success at Sunderland.
Wenger claimed on Friday that Europe is no longer producing forwards like South America and, whether true or not, there are few players in the Premier League capable of producing what Sanchez does.
On 35 minutes, he tried to lift the ball over Heaton from distance, but shot just wide.
For his next effort, five minutes later, Heaton just about managed to parry the ball away.
The goalkeeper, however, was not the only Burnley player who seemed to use his hands in that box. Arsenal had two penalty shouts in the first half, but referee Craig Pawson remained unmoved.
– Ramsey summoned from bench –
Both Santi Cazorla and Danny Welbeck, meanwhile, missed decent opportunities for the home side.
Burnley barely fashioned a chance themselves, but did cause Arsenal one moment of panic in the first half, just before the interval.
The ball broke for Scott Arfield, but just when the winger was about to play a pass to Danny Ings, Mathieu Flamini made a vital touch.
With Arsenal toiling at the other end, Wenger hauled off Mikel Arteta for Aaron Ramsey shortly after half-time.
Despite all of their possession, Arsenal were missing the thrust of players running beyond Sanchez.
But then they almost forced a goal. Fine work from Chambers set up Cazorla, but his shot was blocked by a robust Burnley body.
It was becoming the story of the game, until Sanchez made the difference.
On 70 minutes, Chambers whipped in a cross that looked a little too high for the Chilean, but he showed impressive force to get in between Kieran Trippier and Michael Duff to brilliantly head home.
Within two minutes, the game was settled, as Chambers turned scorer.
A set-piece fell to Welbeck in the box and although his shot was blocked on the line, Chambers was there to poke the rebound home and claim his first senior goal.
Substitute Lukas Podolski almost powered in another two goals, forcing another bloke on the line and then smashing the ball off the post, before Kieran Gibbs teed up Sanchez for his second goal in stoppage time.